Generalized Mersenne Numbers and Prime Number Generation

In this study, Mersenne numbers are generalized to include an infinite set defined by the sequence of odd primes. To implement a straightforward algorithm for generating prime numbers on a mainframe, similar to the construction of Mersenne numbers, this paper uses the transformation of sets of natural numbers into infinite-dimensional row vectors. This results in a compact formulation that describes how natural-number sequences are generated and, like Mersenne primes, includes some prime numbers of its own. However, unlike the Mersenne case, which produces a single sequence containing various primes (including the large known prime), the current process appears to produce infinitely many sequences with the same potential within the broader class of _generalized_ Mersenne numbers. Furthermore, the overall structure of these natural sequences allows for the recursive search for attached primes: knowing one prime enables the generation of another, which can then generate yet another, and so forth. Finally, a brief overview of an alternative approach based on _generalized twin_ Mersenne numbers further expands the scope of this prime-number framework. Several illustrative examples and tables, along with various results, are also included. Two conjectures that encompass all this prime-generating potential are also proposed.